-12. The minor scale-13. Key and transposition-14 and PAGE 1. Quality or timbre of musical sounds-2. Forms assumed by the vibrations, optical method-3. Another optical method. -4. Phonautographic method-5. Laws of harmonics.-6. Quality or timbre of strings and of instruments-7. General THE THEORY OF SOUND. CHAPTER I. 1. PERIODIC MOVEMENTS, VIBRATION-2. SONOROUS VIBRATION—3. VIBRATION OF A BELL-4. VIBRATION OF A TUNING-FORK, GRAPHIC METHOD— 5. VIBRATION OF A STRING-6. OF PLATES AND MEMBRANES-7. VIBRATION OF AIR IN A SOUNDING PIPE-8. METHOD BY THE MANOMETRIC FLAME.-9. CONCLUSION. 1. AMONGST the innumerable forms of motion which exist in nature, the science of Physics pays especial attention to some, to which it assigns great importance. These are those forms of motion in which a body, or part of a body, arrives at an extreme point, remains at rest for a moment, retraces its steps, again takes the road it has already passed over, and continues thus, making regular to-and-fro movements in a determinate line. The pendulum offers us the most simple example of such a periodic movement. Its laws have been determined by Galileo, who discovered that the movement is isochronous-that is to say, that the time in which the to-and-fro movement is executed, is always the same for the same pendulum, be its oscillations large or not. A In other words, if we give to a pendulum at rest a slight impulse, or a strong impulse, the oscillations will be respectively small or large; but for the same pendulum the duration of each oscillation will be always the same; which may be expressed as follows, that the duration of the oscillations is independent of their extent. The law of the isochronism of the pendulum is a very general law in nature. Although it may not be mathematically exact, still it is sufficiently so for the majority of cases which we shall consider. Every periodic to-andfro movement comparable with that of the pendulum is called an oscillation, and if it be smaller and quicker, it is also termed a vibration. For greater clearness we will give the name of simple vibrations to those which exactly follow the laws of the pendulum, which, by the way, are the most simple of all. On the other hand, we will give the name of compound vibrations to those which follow more complex laws. An example will show how vibrations can be more complex. The movement of the pendulum may be thus summed up: When it has arrived at the end of its path, it remains at rest for a moment, and comes back with a constantly-increasing velocity, which becomes a maximum in the vertical position, and then decreases during the second half of its path. For the pendulum, then, the two extreme points of the oscillation correspond to a velocity zero, the middle point to the maximum velocity. An example of compound periodic movement is obtained pen by adding to the already existing oscillation of the dulum some other oscillatory movement. Suppose, for example, that the pendulum rod be flexible and elastic, and that it oscillates on its own account, and let us further suppose the lower heavy part of the pendulum to be an elastic ball, which, being violently impelled when at rest, vibrates like a ball on a billiard-table-that is to say, exhibits successive compressions and expansions. We shall then have three vibratory movements united in the pendulum, which will give a compound movement obviously more complex than the first. Another example of a compound movement is furnished by the ball-player, who throws a ball vertically up in the air, and then sends it up again without allowing it to fall to the ground. Here the movement is different from that of the pendulum. The ball goes up with a velocity decreasing nearly at the same rate as that of the pendulum does (but the velocity decreases according to a different law), comes to rest, and then falls with a constantlyincreasing velocity, and is suddenly stopped and thrown up again by the muscular force of the player, whilst its velocity is fairly high and still increasing, according to the laws which regulate the falling of heavy bodies. In this case, then, contrary to that which takes place in that of the pendulum, the two extreme points of the path correspond -the one to a velocity of zero, the other to a fairly high velocity-in fact, to the highest velocity possible under the circumstances. There exist in nature a very great |